waves2Foam Manual

Abstract

This is the draft version of the waves2Foam Manual. It has 'draft' as a descriptor, since some textual editing might still occur. Nonetheless, the manual is covering the full functionality and replaces previous online documentation.

Full text

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F
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y z
∂ρu
∂t +∇ · ρuuT=−∇p∗+g·(x−xr)∇ρ+∇ · µtot ∇u
ρut
∇= (∂/∂x, ∂/∂y, ∂/∂z)p∗g
x= (x, y, z)xr
µtot
∇ · u= 0
p∗=p−ρg·xp
n
u=nup
up
(1 + Cm)∂
∂t
ρu
n+1
n∇ · ρ
nuuT=−∇p∗+g·(x−xr)∇ρ+1
n∇ · µtot∇u−Fp
CmFp

F
∂F
∂t +∇ · uF+∇ · ur(1 −F)F= 0
urF
ρ=F ρ1+ (1 −F)ρ0µ=F µ1+ (1 −F)µ0
F= 0
F= 1 F= 1 F= 0
F∈[0,1]
∂F
∂t +1
n(∇ · uF+∇ · ur(1 −F)F) = 0
n
φ= (1 −wR)φ+wRφ
wR∈[0,1]

αu
◦
◦
◦

wR
˜wR= 1 −(1 −w∗
R)/max
w∗
R= 1 −wR˜wR
max
wR(σ= 1) = 0 wR(σ= 0) = 1 σ
wR= 1 −exp σp−1
exp 1 −1
p3.5
wR= 1 −σp
p
p
wR=−2˜σ3+ 3˜σ2
˜σ= 1 −σ
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Fp
ρ=au+bukuk2
a b Cm
Cm=γp
1−n
n
γp0.34
a b
a=α(1 −n)3
n2
ν
d2
50
, b =β1−n
n3
1
d50
ν d50

a b
a=α(1 −n)2
n3
ν
d2
50
, b =β1 + 7.5
KC 1−n
n3
1
d50
ν d50
α β KC
KC

3×3 = 9

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◦
−1
0 1
◦σ
◦
1

◦φ3
◦φρ
◦φF3
φ φF
φF
φ φρ
φρ= (ρF=1 −ρF=0)φF+ρF=0 φ
φF
φF=φρ−ρF=0φ
ρF=1 −ρF=0
F= 1
F
q=X
f∈F
φF,f
Sf
kSfk2
q3Sf
φF

φFφ φρ
φF
q

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Hm=H01 + sin 1
N(ωt −k·x)
HmH0
N ω k

h/L h L
m
1
m m
1

η=
N
X
i
aicos(ωit−ki·x+ϕi)
η N ai
i ω kϕ
◦
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◦
aiωi
ωiϕi
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◦

◦
◦
x0t0

0
t= 0

t= 0

x y z
x y z

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◦t= 0
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x
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k−ω
k−
k−ω

u−p


k−ω



KC

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∆f= 1/Tef
Te
100

R(τ) = 1
m0Z−∞
0
S(f) cos 2πf τ f
R(τ)S(f)f τ
m0
Rd(τ)'1
2md,0
N−1
X
n=0
[Sd(fn) cos(2πfnτ) + Sd(fn+1 ) cos(2πfn+1τ)] ∆fn
d∆fn=fn+1 −fnN
η
η(t, x) =
N
X
m=1
amcos (ωmt−km·x+ϕm)
amωmkmx
ϕm[0,2π]
am=p[Sd(fm−1) + Sd(fm)]∆fm−1
ωm=π(fm+fm−1)
τ > 0 cos(2π fnτ)=1
n TrTm01
Tr
fn=nδf0∆fn=δf0n= 0,1...,N
Tr=1
δf0
=N
fN
fN
N fNTr
∆fn= (1 + nαf)δf0
fn=nδf0+αfδf0nn+ 1
2−1=δf0(n+αfAn)
1/(1 −N)< αfαf= 0
cos(2πfnτ) = cos(2πnδf0τ) cos(2παfAnδf0τ)
−sin(2πnδf0τ) sin(2παfAnδf0τ)
n
nδf0τ=K αfAnδf0τ=L
K L τ
K
n
L
An
αf= 1

α
τ Rd= 1
αfTrα= 1/4Tr= 1/δf0Tr= 4/δf0
α= 4 αfδf0
δf0αf
Tr
S
fLfPfU
[fL, fP[
NL=fP−fL
fU−fP (N+ 1)
[fP, fU]NU=N+ 1 −NL
fn=fL+ (fP−fL) sin 2π
4NL
nn= 0,...NL−1
fNL+n=fP+ (fU−fP)1−cos 2π
4NU
n n= 0,...NU
∆f f fL= 0.03
fU= 1.4Tp= 3 N= 200
N= 35
N
N= 200
N= 20
N
Tr'15 Tr
N|Rd|
Tr

0 0.2 0.4 0.6 0.8 1 1.2 1.4
f, [Hz]
10 -4
10 -3
10 -2
10 -1
∆f, [Hz]
A.
0.2 0.25 0.3 0.35 0.4 0.45 0.5
f, [Hz]
0
0.2
0.4
0.6
S, [m2s]
B.
Equidistant
Cosine stretching
∆f f
fL= 0.03 fU= 1.4Tp= 3 N= 200 N= 35
Hm0= 4√m0Tm01 =m0/m1
Tm02 =pm0/m2Tm−10 =m−1/m0N
mii
mi=Z∞
0
fiS(f)f
N
N= 10,000
N= 10,000
N
N
∆t= 0.02 Te= 5,000
TrTr= 730 N= 1,000
N
N N = 100
0.1%
N= 100

0 20 40 60 80 100
-1
-0.5
0
0.5
1
η, [m]
A.
0 20 40 60 80 100
t, [s]
-1
-0.5
0
0.5
1
η, [m]
B.
0 10 20 30 40 50
-1
0
1
R, [-]
A.
0 100 200 300 400 500
t, [s]
-1
0
1
R, [-]
B.
σ σ

10 110 210 310 4
0.98
1
1.02
Rel. Hm0, [-]
A.
10 110 210 310 4
0.98
1
1.02
Rel. Tm01, [-]
B.
10 110 210 310 4
0.98
1
1.02
Rel. Tm02, [-]
C.
10 110 210 310 4
N, [-]
0.98
1
1.02
Rel. Tm−10, [-]
D.
Equidistant
Cosine stretching
N
N= 10,000
η=η0+
N
X
n=1
ancos(ωnt−kn·x+φn)
η η0N ω
tk x φ
η=η0+
N
X
n=1
an[cos(ωnt+φn) cos kn·x−sin(ωnt+φn) sin kn·x]

02468
10 -4
10 -2
10 0
Exceedance
Equidistant
N= 100
02468
10 -4
10 -2
10 0
Exceedance
N= 200
02468
10 -4
10 -2
10 0
Exceedance
N= 500
02468
(H/Hrms)2, (T /Tp)2, [-]
10 -4
10 -2
10 0
Exceedance
N= 1000
02468
10 -4
10 -2
10 0Cosine stretching
N= 100
02468
10 -4
10 -2
10 0
N= 200
02468
10 -4
10 -2
10 0
N= 500
02468
(H/Hrms)2, (T /Tp)2, [-]
10 -4
10 -2
10 0
N= 1000
Rayleigh
Wave height
Wave period
(H/Hrms )2(T/Tp)2
±σ
x
t
2N
z
t x y
◦
x y z

◦
(x, y)z
◦
N= 1
N= 100
10 010 110 210 310 4
10 0
10 1
10 2
10 3
TN/TN=1, [-]
A.
Direct method
Split method
10 010 110 210 310 4
N, [-]
0
10
20
30
Direct over split, [-]
B.
N= 1
5
N= 100 N= 1,000
N2